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I am trying to evaluate the following integral, and I have the answers:

$$\int_C \left(e^z + \frac1z\right)\,\mathrm dz$$

Where $C$ is the lower half of the circle with radius $1$ centre $0$, negatively oriented.

So a primitive is: $e^z + \log z$ and now we must choose a branch cut for the complex log, and they choose:

$$\frac\pi2 \lt \theta \lt \frac{5\pi}{2}$$ I will admit, I don't totally understand the branch cut, beyond the idea that it limits us to a $2\pi$ range, and that it gives us a unique solution. Why not just stick with $-\pi \lt \theta \leq \pi$. Actually that raises a second question, why do they set it to $\lt$ both times, this gives me some insight perhaps: Did they choose this, since there is a line they are excluding? Maybe I just worked it out, they are cutting out the positive imaginary axis for $(0,y)$?

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You have worked it out.

To your first question, remember that $$ \text{log}(z)=\text{log}|z|+i \text{arg}(z) $$ Then you just need to see that, for a branch of the $\text{log}$ to be a single valued function, it must have its argument in a $2\pi$ length interval.

For the branch cut, since the path $C$ does not crosses the positive imaginary axis you will want to pick a branch whose domain is $$ \mathbb{C} \setminus \{ iy : y \geq 0\}, $$ hence the argument of $\text{log}(z)$ must be in $(\pi/2, 5\pi/2)$ or adding a $\pm 2\pi$ shift. Now note that at the begining of the path $C$ ($z=1$), $\text{log}(z)$ will have an argument of $2\pi$ but at the end ($z=-1$) the argument is $\pi$. Thus, parametrizing $C$ as $C(\theta)=e^{i \theta}$, where $\theta$ runs from $2\pi$ to $\pi$, the integral is \begin{align} \int_C \left( e^z + \frac{1}{z}\right)&=e^{C(\pi)}+\text{log}(C(\pi))-e^{C(2\pi)}-\text{log}(C(2\pi))\\ & = e^{-1}+\text{log}(-1)-e^1-\text{log(1)}\\ & = \frac{1-e^2}{e}+\text{log}|1|+i\pi -\text{log}|1|-i2\pi \\ & = \frac{1-e^2}{e}-i\pi \end{align}

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